Flash and strobe are basically the same thing.
Continuous lighting is often referred to by the type of bulbs...incandescent, florescent, halogen etc. They might also be called 'hot lights' etc.
It sounds like you had some of the terms mixed up, but you are asking the right question. When you are considering lights, the first decisions is probably whether you want to get continuous lights or flash/strobe.
Continuous lights have the advantage that what you see is what you get, so you can set everything up to look just right and then shoot it. For the most part,
they are also less expensive than flash/strobe lights.
The downside to continuous lighting is that you need a lot of it in order to get exposures with fast shutter speeds, which you need if you are shooting hand held or shooting moving subjects like people. If you are shooting still objects and can use a tripod, then continuous lighting is fine, but for shooting people, flash/strobes are highly recommended. Also, continuous lighting can get pretty warm for your subjects.
Flash/strobe lighting can be a bit of a mystery because it's typically not 'what you see is what you get'. However, studio strobes do usually have modeling lights which help to give you an idea of what you are doing.
The big advantage of flash lighting is that it's a very quick burst of light which freezes most movement, which usually makes for sharp photos.
So if you are going to go with flash/strobe lighting...there are a few options. Option one is studio strobe lights...typically 'mono-lights'. These lights will usually go on light stands and have to be plugged into a power source. There are some cheap, underpowered kits floating around but a good kit will have plenty of power.
Option two is to use 'hot shoe' flash units. These are much smaller and run off of batteries (usually AA), which makes them a lot more portable than studio lights. Check out
THIS SITE for plenty of info on using these types of lights.
There are pros and cons to each choice...so this is something you need to decide for yourself.